Elastic Problems Research Articles

Spatial behavior in an elastic problem with non-positive definite tensors

Spatial behavior in an elastic problem with non-positive definite tensors

Topology Optimization for Elasticity Problem of Isotropic and Orthotropic Materials Based on the Complex Variable Element-Free Galerkin Method

By using the improved complex variable moving least squares (ICVMLS) approximation to construct the shape function, thus the improved complex variable element-free Galerkin (ICVEFG) method is established for analyzing two types of topology optimization problems including isotropic and orthotropic materials in this study. Solid isotropic microstructure with penalization (SIMP) is utilized with relative density of nodes selected as the design variable, volume fraction set as the constraint and structural flexibility selected as the objective function, thus a mathematical model for topology optimization is developed, and the formula of sensitivity is derived. Four optimization examples of isotropic and orthotropic materials are given, and the feasibility of the ICVEFG method for solving topology optimization problem is demonstrated, in addition, the checkerboard phenomenon can be avoided; it is also shown that the ICVEFG method is more advantageous than the EFG method in solving orthotropic beam as well as isotropic problem with two fixed ends.

Non-intrusive parametric hyper-reduction for nonlinear structural finite element formulations

Model Order Reduction (MOR) is a core technology for the creation of comprehensive executable Digital Twins, since it efficiently reduces the computational burden of high-fidelity models. When dealing with nonlinear structural Finite Element analyses, several Hyper-Reduction (HR) approaches have been developed to reduce the computational cost. Nonetheless, HR approaches are typically intrusive in nature, posing challenges when it comes to integration into existing (commercial) software. Recently, data driven Non-Intrusive MOR methodologies have been proposed. However, these techniques often suffer from overfitting and violate key physics properties, leading to unstable behavior. This work proposes to use Scientific Machine Learning to reintegrate critical stability-preserving physics properties. It introduces a data-driven, physics-augmented, parametric approach that combines Proper Orthogonal Decomposition (POD) with a Partially Input Convex Neural Network (PICNN) architecture. The proposed method effectively reduces the computational burden associated with parametric static nonlinear elastic structural problems while retaining material consistency, hyper-elasticity, and material stability properties in the Reduced Order Model. Numerical validation on several structural models subjected to geometrical and material nonlinearities under static loading conditions demonstrates the effectiveness of the POD-PICNN approach. Additionally, three different sampling strategies have been compared to assess their impact on the method performance. The results emphasize that physics-augmentation is required, as it inherently embeds essential physical constraints into the neural network architecture, ensuring stable and consistent behavior, while highlighting its potential for dynamic and multiphysics applications.

A relaxed Robin-Robin algorithm for solving a Cauchy problem in linear elasticity

A relaxed Robin-Robin algorithm for solving a Cauchy problem in linear elasticity

The mixed displacement method to avoid shear locking in problems in elasticity

AbstractThe mixed displacement (MD) method was initially developed to mitigate geometrical locking effects in beams, plates, and shells with the intention of having intrinsically locking‐free characteristics while using equal‐order interpolation for all degrees of freedom. In other words, it is an unlocking scheme that works independent of the element shape, polynomial order, and discretization scheme. It includes additional degrees of freedom that adhere to a carefully designed differential relation that can be interpreted as a kinematic law, incorporated in a mixed sense. Certain constraints are to be enforced on these additional degrees of freedom to obtain a well‐posed system of equations. In this work, the MD method is extended for problems in solid mechanics. We present the underlying variational formulation, followed by its application to 2D solid elements. Additionally, we showcase an idea to enforce the additional constraints in a general sense. Various numerical examples, within the framework of the finite element method and isogeometric analysis, are outlined to demonstrate the performance of the MD method in the geometrically linear and geometrically nonlinear cases.

Korn and Poincaré-Korn inequalities: A different perspective

We present a concise point of view on the first and the second Korn’s inequality for general exponent p p and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for p = 2 p = 2 , uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent 1 > p > ∞ 1>p>\infty remains the same, the only change being invoking now the q q -Riesz representation theorem (with q q the harmonic conjugate of p p ). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

Approximate kink-kink solutions for the ϕ 6 model in the low-speed limit

In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ 6 model in dimension 1 + 1. We construct a sequence of approximate solutions ( ϕ k ( v , t , x ) ) k ∈ N ⩾ 2 for this model to understand the effects of the collision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ 6 model.

Traction‐associated peridynamic model and non‐uniform discretization simulation of plane axisymmetric problems

AbstractUsing the recently developed traction‐associated peridynamic motion equation (TPME), we investigate the plane axisymmetric problems of elastic deformation. The polar coordinate form of TPME is derived for a plane axisymmetric problem. The transfer function matching with prototype microelastic (PM) model is determined by the inverse method in a one‐dimensional case, and then it is directly extended to plane axisymmetric elasticity problems and plane problems. We promote a strategy to deal with the horizon in the non‐uniform discretization scheme. Based on this strategy, the non‐uniform discretization scheme is used to solve the elastic deformation of a hollow circular plate subjected to uniform inner and outer pressures. Total process from elastic deformation to failure of the hollow circular plate under tension is simulated completely. The results show that TPME does not only need volume and surface corrections, but also can deal with the complex traction boundary conditions conveniently.

Resolution of a nonlinear elasticity problem subject to friction laws

Resolution of a nonlinear elasticity problem subject to friction laws

Data-Efficient Dimensionality Reduction and Surrogate Modeling of High-Dimensional Stress Fields

Abstract Tensor datatypes representing field variables like stress, displacement, velocity, etc., have increasingly become a common occurrence in data-driven modeling and analysis of simulations. Numerous methods [such as convolutional neural networks (CNNs)] exist to address the meta-modeling of field data from simulations. As the complexity of the simulation increases, so does the cost of acquisition, leading to limited data scenarios. Modeling of tensor datatypes under limited data scenarios remains a hindrance for engineering applications. In this article, we introduce a direct image-to-image modeling framework of convolutional autoencoders enhanced by information bottleneck loss function to tackle the tensor data types with limited data. The information bottleneck method penalizes the nuisance information in the latent space while maximizing relevant information making it robust for limited data scenarios. The entire neural network framework is further combined with robust hyperparameter optimization. We perform numerical studies to compare the predictive performance of the proposed method with a dimensionality reduction-based surrogate modeling framework on a representative linear elastic ellipsoidal void problem with uniaxial loading. The data structure focuses on the low-data regime (fewer than 100 data points) and includes the parameterized geometry of the ellipsoidal void as the input and the predicted stress field as the output. The results of the numerical studies show that the information bottleneck approach yields improved overall accuracy and more precise prediction of the extremes of the stress field. Additionally, an in-depth analysis is carried out to elucidate the information compression behavior of the proposed framework.

Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media

Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media

Edge computing resource scheduling method based on container elastic scaling.

Edge computing is a crucial technology to solve the problem of computing resources and bandwidth required for extensive edge data processing, as well as for meeting the real-time demands of applications. Container virtualization technology has become the underlying technical basis for edge computing due to its efficient performance. Because the traditional container scaling strategy has issues such as long response times, low resource utilization, and unpredictable container application loads, this article proposes a method for scheduling edge computing resources based on the elastic scaling of containers. Firstly, a container load prediction model (Trend Enhanced-Temporal Convolutional Network, TE-TCN) is designed based on the temporal convolutional neural network, which features an encoder-decoder structure. The encoder extracts potential temporal relationship features from the historical data of the container load, while the decoder identifies the trend item of the container load through the trend enhancement module. Subsequently, the information extracted by the encoder and decoder is fed into the fully connected layer to facilitate container load prediction using the dual-input ResNet method. Secondly, Markov decision process (MDP) is used to model the elastic expansion problem of containers in multi-objective optimization. Utilizing the prediction outcomes of the TE-TCN load prediction model, a time-varying action space is formulated to address the issue of excessive action space in conventional reinforcement learning. Subsequently, a predictive container scaling strategy based on reinforcement learning is devised to align with the application load patterns in the container environment, enabling adaptation to the surge in traffic generated by the container environment. Finally, the experimental results on the WorldCup98 dataset and the real dataset show that the TE-TCN model can accurately predict the container load change. Experiments in the actual environment demonstrate that the proposed strategy reduces the average response time by 16.2% when the burst load arrives, and increases the average CPU utilization by 44.6% when the jitter load occurs.

Numerical simulation of the creep of a turbine blade made of a monocrystalline alloy

The article is devoted to the creep model of a monocrystalline alloy and the development of a methodology for identifying material parameters based on the results of physical experiments. A finite element analysis of the creep of a gas turbine engine blade was performed. Creep is one of the most dangerous types of deformation in the operating conditions of turbine blades. In the process of studying the problems of assessing the strength of turbine blades of aircraft engines and power plants, special attention should be paid to the study of stress redistribution during creep. The characteristics of the crystallographic structures of modern turbine blades have a very significant influence on the progress of the crack development process during engine operation. To date, turbine blades are manufactured by the method of single-crystal casting. This type of blade material structure is characterized by orthotropic mechanical properties. This study considers the steady-state creep model of an anisotropic heat-resistant single-crystal alloy with cubic symmetry. The authors carried out a numerical simulation of the material parameters using the well-known literary creep properties of single crystals. An algorithm is described that allows you to determine some creep characteristics of single crystals. The parameters of the given ratios can be obtained after conducting direct experiments, or based on micromechanical analysis, using the example of composite materials. The authors calculated the creep constants of a typical heat-resistant monocrystalline alloy as a result of the approximation of its creep curves, which were obtained experimentally. Based on the Norton-Bailey equation and using the Maple Release 2021.0 calculation complex, a graph of the dependence of the rate of creep deformation on the level of load applied to the material was plotted, and the minimum rate of deformation and creep constants were also determined. The results of the calculations were used for finite-element simulation of creep on the example of a solid-state model of a high-pressure turbine blade. Several series of calculations were carried out on the basis of the ANSYS Workbench complex, in particular, the calculation of the elastic problem when the blade is loaded by centrifugal forces, as well as the accumulation of creep deformations at different exposure times. Graphs of the change in equivalent stresses and creep deformations as a function of time are plotted.

Nonlinear mechanics of phase-change-induced accretion

In this paper, we formulate a continuum theory of solidification within the context of finite-strain coupled thermoelasticity. We aim to fill a gap in the existing literature, as the existing studies on solidification typically decouple the thermal problem (the classical Stefan’s problem) from the elasticity problem, and often limit themselves to linear elasticity with small strains. Treating solidification as an accretion problem, with the growth velocity correlated with the jump in the heat flux across the boundary, it presents an initial boundary-value problem (IBVP) over a domain whose boundary location is a priori unknown. This IBVP is solved numerically for the specific example of radially inward solidification in a spherical container. Several parametric studies are conducted to compare the numerical results with the rigid cases in the literature and gain insights into the role of elastic deformations in solidification.

Implementation of the Zienkiewicz–Pande Model into a Four‐Dimensional Lattice Spring Model for Plasticity and Fracture

ABSTRACTPlasticity and fracture problems have always been hot topics in numerical methods. In this work, a universal implementation procedure for the elasto‐plastic constitutive model is developed in the four‐dimensional lattice spring model (4D‐LSM), in which the Jaumann stress rate is incorporated to exclude the influence of the rigid rotation in the particle stress, expanding the ability of 4D‐LSM to deal with large elastic deformation problems by its own to large plastic deformation problems. As an example, the Zienkiewicz–Pande (ZP) constitutive model is implemented. Several numerical examples are carried out to check the performance of the implemented model. Through a comparison with analytical solutions, available experimental data, and other numerical results, the stability of the developed plastic framework and the correctness of the stress calculation scheme are verified. Meanwhile, numerical results show that the developed code is capable of solving elasto‐plastic large deformation problems. With the advantage of 4D‐LSM in handling fracture problems, the ability of the embedded model to solve plastic fracture problems is verified with a simple maximum deformation failure criterion.

Level-set based shape optimization for plane elastic structures using radial basis functions and Hilbertian descent direction

Structural optimization problems are often associated with the so-called shape functionals depending on a shape through its geometry and the state being a solution of given partial differential equation. In such a framework it is convenient to work with the gradient-like method based on a concept of a shape derivative and level set method. The key idea of level set method is to represent the structural boundary with zero level set of given function (level set function—LSF). Now, changing the shape of a structure under optimization is equivalent to transport the LSF in such a direction that ensures decreasing the value of the objective functional. To this end, we make use of coercive bilinear form taken from the weak formulation of elasticity problem to obtain descent direction at each iteration. This descent direction is a solution of an additional variational problem, involving the bilinear form mentioned above and the volumetric expression of the shape derivative plays the role of a linear form. In this paper, we combine level set method with radial basis functions (RBFs) used to approximate LSF. We focus on the so-called multiquadric RBFs, but other classes of RBFs are also briefly considered. This eventually leads to transformation of partial differential equation (linear transport equation governing the evolution of shapes) to a system of linear ordinary differential equations which admits analytical formula for the solution. We apply our method to compliance minimization of a cantilever problem as well as to total potential energy minimization of a structure with kinematic loading. To run all the numerical experiments, we wrote our own code in Wolfram Mathematica environment.

Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

Existence results for the time-incremental elastic contact problem with Coulomb friction in 2D

In this paper, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini–Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray–Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray–Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini–Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.

A non-intrusive multiscale framework for 2D analysis of local features by GFEM — A thorough parameter investigation

This work comprehensively investigates key parameters associated with a recently proposed non-intrusive coupling strategy for multiscale structural problems. The IGL-GFEMgl combines the Iterative Global Local Method and the Generalized Finite Element Method with global–local enrichment, GFEMgl. Different scales of the problem are solved using distinct finite element codes: the commercial software Abaqus and a research in-house code. An Iterative Global–Local non-intrusive algorithm is employed to couple the solutions provided by the two solvers, with the process accelerated by Aitken’s relaxation. Slight modifications have been introduced, and the resulting accuracy and computational performance are discussed using numerical examples. The problems investigated explore the coupling strategy within the context of 2D linear elastic problems, which include voids and crack propagation described at the local scale solved by the in-house code. A noteworthy trade-off between reducing iterations and increasing the time to solve the local problems is observed. Despite the high accuracy achieved, the two versions of the coupling strategy, namely the monolithic and staggered algorithms, exhibit different computational performances when the GFEMgl parameters, such as the number of global–local cycles and the size of the buffer zone, are evaluated for the crack propagation simulation.

Mixed displacement–pressure formulations and suitable finite elements for multimaterial problems with compressible and incompressible models

Multimaterial problems in linear and nonlinear elasticity are some of the least explored using mixed finite element formulations with higher-order elements. The fundamental issue in adapting the mixed displacement–pressure formulations with linear and higher-order continuous elements for the pressure field is their inability to capture pressure and stress jumps across material interfaces. In this paper, for the first time in literature, we perform comprehensive studies of multimaterial problems in elasticity consisting of compressible and incompressible material models using the mixed displacement–pressure formulation to assess the performance of different element types in accurately resolving pressure fields within the domains and pressure jumps across material interfaces. In particular, inf–sup stable displacement–pressure combinations with element-wise discontinuous pressure for triangular and tetrahedral elements are considered and their performance is assessed along with the Q1/P0 element and Taylor–Hood elements using several numerical examples. The results show that Taylor–Hood elements fail to capture the stress jumps due to the continuity of DOFs across elements, the Crouzeix–Raviart (P2b/P1dc) element yields substantially poor pressure fields despite a significant increase in pressure degrees of freedom and that the P3/P1dc element produces superior quality results fields when compared with the P2b/P1dc element.